Properties

Label 66270.k
Number of curves $8$
Conductor $66270$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 66270.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
66270.k1 66270m7 \([1, 0, 1, -11781748, 15564503978]\) \(16778985534208729/81000\) \(873116441649000\) \([2]\) \(2543616\) \(2.4892\)  
66270.k2 66270m8 \([1, 0, 1, -1001828, 52446506]\) \(10316097499609/5859375000\) \(63159464818359375000\) \([2]\) \(2543616\) \(2.4892\)  
66270.k3 66270m6 \([1, 0, 1, -736748, 242879978]\) \(4102915888729/9000000\) \(97012937961000000\) \([2, 2]\) \(1271808\) \(2.1427\)  
66270.k4 66270m5 \([1, 0, 1, -637343, -195893692]\) \(2656166199049/33750\) \(363798517353750\) \([2]\) \(847872\) \(1.9399\)  
66270.k5 66270m4 \([1, 0, 1, -151363, 19510316]\) \(35578826569/5314410\) \(57285169736590890\) \([2]\) \(847872\) \(1.9399\)  
66270.k6 66270m2 \([1, 0, 1, -40913, -2888944]\) \(702595369/72900\) \(785804797484100\) \([2, 2]\) \(423936\) \(1.5934\)  
66270.k7 66270m3 \([1, 0, 1, -29868, 6499306]\) \(-273359449/1536000\) \(-16556874745344000\) \([2]\) \(635904\) \(1.7961\)  
66270.k8 66270m1 \([1, 0, 1, 3267, -220472]\) \(357911/2160\) \(-23283105110640\) \([2]\) \(211968\) \(1.2468\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 66270.k have rank \(0\).

Complex multiplication

The elliptic curves in class 66270.k do not have complex multiplication.

Modular form 66270.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} + q^{12} - 2 q^{13} + 4 q^{14} + q^{15} + q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.